A theoretical and numerical model of ship motions in heavy seas

She wanted care in loading and handling,

and no one knew exactly how much care

would be enough. Such are the

imperfections of mere men!

Joseph Conrad, The Nigger of the "Narcissus"

1.0 INTRODUCTION

Those words of Conrad, written in

1896, strike at the heart of the predicament faced by a seaman who, to

overcome the power of the sea, entrusts himself to the seaworthiness of his ship. She, the ship personified in the presence of ultimate danger, and her all important motions become the ultimate enigma.

In 1898 the first comprehensive method for the calculation of

hydrodyn.mic loads and ship motions in waves was published, (Krylov, 1898). The method was based ori a simple physical assumption that the presence of the ship does not change the pressure distribution

TECHfUSCHE UNIVERSiTET

Laboratorium voor

Scheepshydromechagj

Archief

Mekelweg 2,2628 CD Deift

TeLi 015.. 786873. Fax: 015.781833

Paper No. 11 - 10:45 am., November 15, 1991

A Theoretical and Numerical Model of Ship Motions in Heavy Seas Jacek S. Pawlowski (N) and Don W. Bass (N)

The paper provides a description of a theoretical model of ship

interaction with heavy seas. In the model the non-linear scattering problem of wave-ship interaction is formulated and solved in the time domain on the basis

of the weak scatterer hypothesis and with the use of the method of modal

potentials. This results in a non-linear scattering problem with respect to the oncoming wave potential (and, therefore, amplitude), which is reduced to a quasi-linear form by means of the modal potentials method. In this way the conceptual framework of the practising naval architect is expanded to cover the non-linear and transient flow phenomena related to ship motions in waves. On that basis, an efficient numerical model is developed which utilizes the latest achievements in the development of methods for the computation of radiation

potentials and forces, but does not require a super-computer processing

capability. The example computations confirm the applicability and efficiency of the models and the validity of the weak scatterer hypothesis.

in the propagating wave. The assumption is now known generally as the Froude-Krylov hypothesis. The progress made since, in the modelling of ship interaction with waves, consists mainly

in the development of techniques for computing the very disturbances discarded by the hypothesis. Reviews of this development can be found in (Newman, 1983) and (Hutchison, 1990) - Most of the current knowledge of the disturbance phenomena is founded on the assumption of small oncoming wave amplitudes and small ship motions, and of small disturbances of the oncoming wave, which are induced

by the ship. This allows a linear

decomposition of the problem into the

radiation (disturbance induced in calm water by a hull oscillating about a

stationary or steadily advancing mean position) and diffraction (disturbance induced in the wave by a stationary or steadily advancing hull) problems, which are solved separately. The solutions to those problems can be obtained either in the frequency domain, see e.g.

(Faltinsen and Michelsen, 1974), (Hogben and Standing, 1974), (Chang, 1977), and (Inglis and Price, 1982), or in the time domain,(Liapis, 1986), (Beck and Liapis,

1987) , (King, 1987) , using so called panel methods. The solutions are then expressed by means of Green's functions which satisfy appropriate linear free surface conditions. Among the panel methods, a high level of robustness and reliability has recently been achieved in

solving the linear frequency domain problems without forward speed, (Newman and Sclavounos, 1988) . Solutions to the other problems must still be considered a

subject of research rather than routine application, (Newman, 1991) . A similar

comment can be extended to methods aimed at the inclusion of non-linear effects in the frequency domain solution with zero forward speed, e.g. (Ogilvie, 1983), (Lee, Newman, Kirn and lue, 1991) , and (Pawlowski, 1991) . Time domain methods

offer an advantage by in principle allowing an arbitrary motion of the hull, and therefore making it possible to abandon the assumption of small ship motions, and the distinction between the cases with and without forward speed. However, the assumption of small oncoming and disturbance wave amplitudes must be maintained. In addition, a direct application of a time domain panel method, to the prediction of large ship motions in waves, constitutes at present a task of daring computational complexity and cost, (Lin and lue, 1990), and (Magee, 1991)

In this paper, by presenting formulae (derived by one of us and developed into a computer program by the other), and examples of computations of ship loads and motions in waves, we aim at contributing to the knowledge of "how much care would be enough". The formulae are to provide a relatively simple, rational conceptual framework for analyzing the problem, as well as a basis for numerical evaluations. The computer program is to be perceived as the tool of

B Cß CG e e F F5 F

f

f

J'

g

h L N P PS Pu Pw R S1 breadth block coefficient center of gravity

unit basis vectors directed along axes X, i=1,2,3

unit basis vectors of the axes system fixed on the ship,

i=l, 2, 3

resultant hydrodynamic forc resultant scattering force resultant Froude-Krylov force resultant scattering force in the reference configuration generalized components of i th modal scattering force,

il,2,..,m, j=l,2,..,6 acceleration due to gravity wave height

length between perpendiculars length on the waterline unit normal vector on S,, directed into the hull pressure

scattering pressure pressure induced by the forward motion of the ship oncoming wave pressure tensor of rotation

free surface of water

the analysis, as well as the source of numerical evaluations.

In the theoretical model described below, the oncoming wave which interacts with the ship, is assumed to be high and steep, so that the quantities proportional to the square of its height cannot be neglected. The motions of the

ship, induced by the wave, are also considered to be large, of a magnitude proportional to the wave height. Instead

of adopting the Froude-Krylov hypothesis, current formulations of the problem of interaction of a ship with an oncoming wave assume that the disturbance caused in the oncoming wave flow by the presence of the ship hull is proportional to the wave height. The theoretical model used here is based on a different key assumption. The disturbance induced by

the moving ship

in the wave flow is

considered to be of a smaller magnitude than the wave flow quantities which are proportional to the wave height, but at least of the same magnitude as the wave flow quantities proportional to the square of the wave height. This assumption, explained here in simple terms, is called the weak scatterer hypothesis, (Pawlowski, in preparation). It is an assumption about the physics of ship motions in waves, as are the Froude-Krylov hypothesis and the currently used paradigm. The weak scatterer hypothesis applies when the ship moves compliantly with the waves, and this usually happens for a free floating ship which operates in steep waves of a length and height comparable to the ship's dimensions. The Nomenclature So SW S,, ( ) T t U u us uu uw va X X

wetted surface of the hull in the reference configuration

instantaneous wetted surface of the hull

instantaneous wetted surface of the hull determined by the elevation of the oncoming wave draft

time variable

mean horizontal speed of the ship

water velocity

water velocity due to the scattering flow

water velocity due to the forward motion of the ship water velocity of the oncoming wave

velocity relative to the reference configuration of the ship

instantaneous normal scattering velocity on the hull surface

radius vector in the reference configuration

presentation of the theoretical model based on the weak scatterer hypothesis is self-contained. However, the model can also be considered as one of several possible formulations in a broader scheme of perturbation type models. For this aspect of the model presented here,

references are made to (Pawlowskl, in preparation).

It should be noticed that a weak

scatterer assumption was used before in

(Newman. 1970) as reported in (Salvesen. 1974), in an investigation of hydrodynamic loads ori submerged bodies, and in (Salvesen, 1974) to simplify expressions for steady second-order

hydrodynamic

loads induced on conventional surface ships. The latter

assumption was made in the context of

"strip theory", (Salvesen, Tuck and Faltinsen, 1970), and can be justified

for slender ships operating at normal

speeds in head and bow waves. The present hypothesis is not limited in those ways. Instead, it depends on the compliant motion of the ship in waves and is applicable to the modelling of large ship motions.

The weak scatterer hypothesis leads to major simplifications of the general problem of ship interaction with waves. However, a sufficiently accurate and efficient evaluation of the flow disturbance induced in the wave by the presence of the moving ship, requires the use of the modal potentials method,

(Pawlowski, 1982) and (Pawlowski, Bass and Grochowalski, 1988). By means of that method the flow disturbance is expressed

by a finite number of modal velocity potentials with unknown time dependent amplitudes which are evaluated in the

X' XI rCG rCGS rcGw Yo e i. 77 77s 71U

coordinate axes fixed in the reference configuration of the

ship, il,2,3

image in S, of point in S amplitudes of modal

potentials, i=l,2,..m

corrected for instantaneous submergence of the hull, i=1,2,..,m

resultant hydrodynamic moment (about CG)

resultant scattering moment (about CG)

resultant Froude-Krylov moment (about CG)

resultant scattering moment in the reference configuration perturbation parameter

elevation of the oncoming wave elevation of the free surface elevation of the free surface due to

elevation of the free surface due to

course of a time domain simulation of the hydrodynamic loads and ship motions. The amplitudes determine corresponding modal potentials using appropriately defined, pre-selected potential influence

functions (instantaneous response and memory potentials). The -rethod does not require any distinction between the radiation and diffraction phenomena. The flow disturbance is represented by a

single scattering velocity potential

obtained by adding together the modal

potentials. The solution takes into

account the instantaneous kinematics of the ship's motion and wave flow, and the memory effects resulting from the propagation of the scattered waves. As a result, the scattering potential and scattering loads are expressed, in terms of the amplitudes of modal potentials, by quasi-linear formulae which involve convolution integrals in time. The important viscous flow phenomena related to roll damping, and lift and drag contributions to sway force and yaw moment, are included using appropriately

adapted,

known

semi-empirical

expressions, (Himeno, 1981) and (Crane, Eda and Landsburg, 1989). The scattering and viscous forces and moments, computed for the six rigid body modes of motion, are applied together with rudder forces and a propulsive force, in the general equations of rigid body motion. The motions of the ship which is assumed to advance with a constant mean forward speed are simulated by a time stepping procedure.

In addition to providing a unified solution to the scattering problem, and facilitating the solution in the

non-linear formulation, the method of modal

and K'(t) vectors of modal memory forces

and K°' (t) vectors of modal memory moments

modal memory potentials, i=1,2,..,rn

wave length

and ' vectors of modal added mass, frequency independent, i=1,2,..,m

and vectors of modal added moments, frequency

independent, i=1,2,..,m water density

velocity potential

scattering velocity potential velocity potential due to the

forward motion of the ship velocity potential of the

oncoming wave

modal scattering potentials, í=l,2,. .m

normal shape functions,

i=i2,..,m

modal instantaneous response potentials, i=l2,..,m

potentials makes feasible the performance

of routine time domain simulations of

ship motions in the design office environment.

Since the potential and

load influence functions of modal amplitudes can be computed in advance, in the course of a tine domain simulation only the computation of modal amplitudes is required in order to obtain the time domain solution to the scattering problem. The modal amplitudes are determined from simple kinematic relations derived from the condition of the impermeability of the hull surface, and therefore the use of the method leads to an important economy of computational effort. As a result, once an appropriate set of load influence functions (generalized added masses and memory forces) is pre-computed for a given ship loading condition and forward speed, time domain simulations can be performed effectively at this loading condition and forward speed, for arbitrary wave conditions, and course angles, on inexpensive computer workstations. In

the application of the method presented here, this flexibility extends to forward speed since modal potentials are determined without taking into account forward speed effects on the free surface.

An implementation of the general theoretical model depends on a number of numerical algorithms which must be chosen to perform the individual computational tasks. One of the most crucial choices to be made is the selection of the algorithm used to generate the modal potentials. A detailed discussion of this topic is

presented below.

All of these choices

have an effect on the final performance of the numerical model, and many of them may be made only after sufficient experience is gained. The development of

"strip theory", and of the frequency domain panel methods, provide examples of the time and effort required to establish

a refined form and the range of

application of a complex hydro-numeric model. The situation is more complicated

for a non-linear time domain model cf

ship motions, owing to the complexity and

required quality of model tests which

could be used to evaluate in detail the adequacy of the theoretical formulation and numerical algorithm. It is fair to state that a comprehensive set of

experimental data entirely suitable for

such a purpose, with respect to large

ship motions in waves, does not exist at present. This is partly due to the new quality present in the use of a hydro-numeric model of this type, which follows from the high degree of required control over the parameters of a simulation. A comparable control cannot be achieved by standard means in the corresponding model tests. This introduces a significant margin of uncertainty in comparisons of simulations of ship motions in waves with experimental data, for the most interesting regimes of operation and wave

conditions.

For the reasons explained

above the comparisons of experimental data with results of numerical simulations are considered in the present paper as examples of computations. Although the experience gathered through the application of the earlier implementations of the presented theoretical model, (Pawlowski and Wishahy, 1987), (Pawlowski, Bass and Grochowalski, 1988), (Wishahy and Pawlowski, 1989) and (Pawlowski and Bass, l990,a), gives additional indications of its usefulness.

The examples of computations

presented here consist of time domain

simulations of fluid loads and ship motions for two hulls. One is a

conventional Series 60 model, (Lewis and Numata, 1960), and the other is a low LIB stern trawler, (Grochowalski, 1989). For

the Series 60 model the simulations are performed in regular waves of small amplitude, in fully captive and free running conditions. For the trawler the simulations are carried out in partly captive and free running conditions in periodic, steep (close to breaking and breaking) waves. The choice of the examples and the comparisons between the experimental and simulated records, are influenced by the available experimental data. However, within this restriction, the comparisons give a representative indication of the capabilities and limitations of the theoretical model and of its present numerical implementation. In particular, the comparison of

simulated load and notion records with

their

experimentally

obtained

counterparts for the trawler confirm the applicability of the models presented, and the validity of the weak scatterer hypothesis.

2.0

THE HYDRODYNAI4IC MODEL

2.1

The kinematics of ship motion and water flou; the weak scatter hypothesis At every point in time the interaction of a ship hull with the surrounding water takes place at their interface, the instantaneous wetted surface of the hull. The interaction involves quantities of kinematic and kinetic kind. The kinematic interaction follows from the necessary compatibility between the motion of the wetted surface and the flow of water. With the assumption of ideal fluid flow, this compatibility is expressed entirely by the impermeability condition which requires that at any point of the wetted surface the speed of the water particles in the direction normal to the surface be equal to the speed of the surface itself

in the same direction. This condition is of fundamental importance in what follows, as it totally determines the flow induced by the presence of the hull. Furthermore, some of the key assumptions and techniques employed in the

theoretical and numerical models to be discussed are closely related to an

understanding and application of that condition.

The ship is considered to be advancing with a mean forward speed U. It is therefore convenient to describe the ship's motion in a right-handed Cartesian system of reference (see Figure

1) which moves with corresponding horizontal velocity U in the direction of its axis X1, relative to a point fixed on the earth. Velocities with respect to that point will be called absolute. The axis X of the system is directed vertically upwards, and the axis X2

completes the system. Axes X1 and X, are placed at the undisturbed water surface. The unit basis vectors along the respective axes are denoted by a, with

i=l,2,3. Velocities observed in that moving system of reference will be called

relative. In the absence of an oncoming wave or other disturbances to the ship's motion, the hull assumes a steady position in the advancing system of

coordinate axes. That position will be called the reference configuration of the ship. In the reference configuration, axes X1 and X3 are in the assumed plane of

lateral symmetry of the hull.

'ig.i Reference frames and mappings from the reference to an instantaneous

configuration of the hull surface Another Cartesian system of reference is fixed on the ship. The unit basis vectors directed along the respective axes of this system are denoted by ', i=l,2,3. When the ship is in its reference configuration the axes of the two systems of reference, which have the same indices, are parallel. The instantaneous relation between the two systems of basis vectors is expressed by means of the tensor of rotation R, as

explained in Appendix 1 which gives an

account of rigid body dynamics relevant to the present considerations. The tensor can be understood as an operator which, according to the ship's motion, rotates basis vectors ' from their positions a

in the reference configuration of the

ship to the instantaneous positions. Symbolically this is expressed by the

formula:

= R i = 1, 2, 3. (1)

The origin of the primed system of

reference is located at the center of

gravity of the ship, and is denoted by

CG.

The impermeability condition which must be satisfied on the instantaneous

wetted surface of the hull, S, can be

written as:

onS

(2)

with ü signifying the absolute velocity of water particles adjacent to the hull, denoting the relative velocity of the

hull,

and ñ

denoting the unit normal vector to the surface, directed into the hull. The dot means the scalar product of vectors. Water velocity i1 may be considered to be the sum of three velocities:

where Q is the velocity generated by the flow of an oncoming wave, and îL is the velocity induced by the forward motion of the ship. Velocity û is induced by the presence of the hull in the wave flow, and includes the interaction effects between the flow generated by the undisturbed oncoming wave (ü) and the flow due to the steady forward motion of the ship in its reference configuration (l31,). Such a decomposition of the velocity 3 involves expansions of the

independently defined velocity fields u, and LT beyond their original domains of

definition, by means of Taylor's series. Using (3), the impermeability condition

(2) is rewritten in the form:

= () .

+ (-)

, on Si,.

(4)

and it follows from the definition of velocity ü that it satisfies the

impermeability condition:

(U-)N= O, on S0

(5)

where S, is the wetted surface in the reference configuration of the ship, and Ñ denotes the unit vector normal to S0.

The flow of the oncoming wave constitutes a disturbance of the calm water condition, of magnitude with respect to an appropriate scale quantity. Here it is convenient to adopt L,, the length of the ship on the calm waterline, as the scale of length, and

_(L7/g)'

as the scale of time, with g denoting acceleration due to gravity. Therefore e3

speeds are scaled by (Lg)'2, i.e. they are represented by the corresponding Froude numbers. With that convention the magnitude of the wave disturbance is indicated by = O(e) . The ship's

motion about its reference configuration

is assumed to be of the same order of

magnitude as the oncoming wave, therefore:

= O(e) , = o(e)

=0(e) and = O(e)

with denoting the instantaneous radius vector, in the X system of reference, of the point of the ship with radius vector X in the reference configuration, see

Figure 1.

In addition, it is supposed that either as a result of the forward speed of the ship being sufficiently small or owing to the slenderness of the ship's

hull, or as a consequence of an appropriate combination of both of those factors, the magnitude of velocity field is significantly smaller than that of velocity field In other words, in the adopted scale, _t» represents a quantity of a smaller order of magnitude than

o(e) . This premise is written as:

= o(e) (7)

and is satisfied in particular if = Q(2) . On the basis of assumption (7), the impermeability condition (4) can be approximated by the expression:

n=

_ons

(8)

which contains all quantities of the orders of magnitude _O(e) and O(e) present in (4), but not all quantities present in (4), which are of the order of magnitude o(e2).

Following (Pawlowski, in prepar-ation), after the above preliminary considerations, the interaction of the ship with the oncoming waves is assumed

tc

be

such that the disturbar,c of he

wave flow by the presence of the ship's hull is significantly smaller than the disturbance of the calm water condition represented by the oncoming wave, but not significantly smaller than the non-linear flow effects in the oncoming wave. In

short, this is expressed by saying that the ship is a weak scatterer. The weak scatterer hypothesis can be conveniently written in the form resulting from (8):

on S, (9a)

vn = (-)

+ U . (9b)

v1 = a(e) and e2 = Q(v) (9c)

(6)

and as a consequence of the hypothesis:

ia-sI = o (e) ande2

= o()

(10) In (9c) and (10) the expressions 2

0(v) and ¿2

. O(,!)

_{mean that v and jj} are not significantly smaller than .

Several comments may elucidate the meaning of the hypothesis. Firstly, the hypothesis applies to the physics of ship motion in waves: it implies that the ship moves compliantly with the waves. The

meaning of the required compliance of

motion is illustrated by considering a ship fixed in its reference configuration in waves. For such a ship y = n - N = 0, and therefore it follows from (8) that in general v, and

_ISI

must be of the same order of magnitude as

_{JW1 ,}

and the weak scatterer hypothesis is not applicable. Secondly, the hypothesis does not involve a mathematical contradiction, since the sum of quantities of O(e) on the right-hand side of (9b) can produce a

significantly smaller quantity v of the order of magnitude O(e), which however is not significantly smaller than the

non-linear wave effects of O(e2) . This

underlines the importance of evaluating

ü in (9b) with sufficient accuracy on the instantaneous wetted surface

S.

Therefore, in order to achieve an

adequate simulation of fluid loads and ship motions, an accurate determination of the instantaneous position of the ship in waves is necessary. This aspect of the resulting theory is confirmed by numerical simulations. Thirdly, a hierarchy of theories of ship motions in waves, of increasing order of consistency, can be developed on the basis of the hypothesis, _(Pawlowski, in preparation) . The group of lowest (first) order theories which contain all terms of the order of magnitude O(E), but do not contain all terms larger than or of the order of magnitude O(e2), includes non-linear formulations such as presented in (Oakley, Paulling and Wood, 1974) and (de Kat and Paulling, 1989). The theory applied here belongs to

the group of

second order theories, which contain all terms of the order of magnitude O(E) to O(e2), including all terms of the order of magnitude 0(62). _{It should be observed}

that for any such theory its order of

consistency depends on the complete inclusion of terms of the corresponding orders of magnitude. Therefore, terms of

a smaller order of magnitude than required by the order of consistency can be included in a consistent theory if it is convenient, and such an approach is adopted here.

The free surface of water, S1, is a material surface consisting of water particles and consequently the evolution of its geometry in time must conform to the motion of the particles. This leads to a kinematic condition which by analogy to (4) is expressed as:

-) .

(U-)

T, on Sf

with , denoting the single valued

elevation of the free surface, and a/öt signifying differentiation with respect to time in the X system of reference. For

the oncoming wave (00, 0O) condition

(11) is reduced to:

-

u-)c_.vc

= o, n ₌

(12) with signifying the free surface elevation due to the oncoming wave. For the steady wave pattern generated by the ship advancing in its reference configuration (ü-O, ü,,=O,

(a/at)=O)

condition (11) leads to:

QnX3=O

(13) where _,, denotes the free surface elevation of the steady wave pattern. Taking into account (12) and (13), condition (11) gives:

Qn X30

(14)

with denoting the free surface elevation induced by the scattering flow. It is seen that (12), (13) and (14) are the well known kinematic conditions on the free surface for respectively: a

propagating wave (non-linear), a steadily advancing wave pattern (linear), and a propagating wave (linear).

With the assumption of irrotational flow the velocity fields discussed above can be expressed by means of their corresponding velocity potentials

,

,

and

,

such that:

ii=V,

_:i=v'

and :i=v

(iSa) with:

=

+ Z', +'

(l5b)

The continuity equation for the flow,

(Newman, 1980), then takes the form of Laplace's equation for potential :

(16)

It follows that Laplace's equation is

also satisfied by each potential on the right-hand side of (l5b). In addition, relations (15a) imply that the potentials can be assigned the same orders of magnitude as their corresponding velocity fields.

2.2

The dynamics of water flow; hydro-dynamic

loads

With the assumption of an

irrotational flow of ideal fluid, the kinetics of the fluid flow are determined by Bernoulli's equation:

P

= -

-

U--at ax1) +

uu

+ X3} (17)

which determines the pressure field p in the water, (Newman, 1980) , if the velocity potential is known, with p denoting the density of water. On the free surface, pressure p can be considered as a constant for the present application, and as usual it is

convenient to take this constant as 0. Therefore, from (17):

(a

at on

s

(18) By a reasoning analogous to the one used in the derivation of the kinematic free surface conditions (12), (13) and (14)

from condition (11), the following kinetic conditions on the free surface are obtained from equation (18):

(-- _{+ + gC} =

o,

(19) on X3 =

for the oncoming wave potential,

-

gi=0,

on X3 =0 (20)

for the potential of the velocity field induced by the steady forward motion of the ship, and:

(

- U_-_- +

ax) s

gi5=0,

on X3=ò (21)

for the scattering potential.

By combining equations (12) and (19) the dynamic free surface condition for potential , is obtained in the form:

(a

a

-+ - .V ( _{w'w) =} 0, on X3 = Ç

This is the full non-linear condition for the potential of a propagating wave, (Newman, 1980), and the last term on the

left-hand side may be neglected as its order of magnitude is Q(3). Similarly, from formulae (13) and (20) the dynamic free surface condition for potential

is derived:

+ _{= 0,} on X3 = 0 (23)

which is Kelvin's free surface condition used in linear predictions of ship wave resistance; e.g. (Baar and Price, 1988). In turn, equations (14) and (21) result in the linear dynamic free surface condition for scattering potential

[ (i

-+ = 0, g, = o

ax, j

(24) It follows from the above discussion that in order to determine the flow of water around the moving ship, the three velocity potentials , u and must be

known.

According to the problem under

consideration, velocity potential which represents the oncoming undisturbed wave is given. In addition, the flow effects which are non-linear with respect to the elevation of the oncoming wave are assumed to be significant and therefore the oncoming wave representation must include at least terms of orders of magnitude o(e) and

0(62)

. From equations (5) and (23), together with Laplace's equation (16), it is concluded that

potential , is the solution to the so

called Neumann-Kelvin problem

of wave

resistance; e.g. (Baar and Price, 1988). However, , is not involved in relations

(9) and (24), and therefore it does not

influence the scattering potential $. It is also shown below that, within the adopted level of consistency, does not contribute to the unsteady hydrodynamic forces exerted on the ship. As a result knowledge of is not necessary for solving the unsteady ship motion problem with the second order consistency. Consequently, the solution to the problem depends on determining the scattering potential, 4's, which must satisfy Laplace' s equation, impermeability condition (9), free surface condition

(24) and appropriate initial or causal conditions.

The dynamic interaction between the ship and surrounding water is defined by the pressure distribution on the instantaneous wetted surface of the ship, S. This pressure distribution is given

by Bernoulli's equation (17) if the right-hand side of the equation is evaluated on SW. The generalized hydrodynamic forces exerted on the ship

in the modes of rigid body motion are

provided by formulae:

X dS (25b)

where denotes _the resultant hydrcdynamic force,and rCQ the resultant hydrodynamic moment about the centre of gravity CG, and x _{denotes vector}

multiplication.

From equation (17), pressure p in formulae (25) is expressed as the sum of three components: p = Pw + Pu + Ps (26a) where: =

p[(

-+

ww +

x3]

at

ax1) w ₂ (26b)

is the pressure in the undisturbed oncoming wave,

Pu = P

(26c)

is the pressure disturbance induced by the steady forward motion of the ship,

and:

PS =

_{P (-}

-

(26d)

is the scattering pressure. The resultant force and moment which correspond to _Pw' are (within the adopted approximation) obtained from expressions:

=

f(c)Pw dS

(27a)

and

rcw

= f5

Pw(cc)

X

ds

(27b) with SW(fl signifying the instantaneous wetted surface with boundary determined on the moving hull by the free surface

of

the undisturbed oncoming wave. Therefore formulae (27) represent so called Froude-Krylov hydrodynamic force and moment.

A different approach must be taken in deterxnininq the hydrodynamic forces due to pressures _Pu and p5. Following (Pawlowski, in preparation), it is assumed that a one-to-one mapping X1 =

(,t)

between points 5 in SW and points X1

in S0 can be established see Figure 1) so that under the mapping S0 is the image of SW. Using this mapping it is possible to transform the integrals over S,, into integrals over S. The transformations are performed for both pressure fields Pu and p5 in the same way, and the result may be written as:

(Pu' PS) ndS = R f(u. ps,) NdS

p5) X

ndS

= R f(P, P5)

(LÏCO)

X

NdS

On

the right-hand sides

of

relations

(28), Pu denotes the pressure determined

on S0 from the solution to the

Neumann-Kelvin problem, whereas Ps signifies the

pressure obtained on S, from the solution

to the scattering problem appropriately

defined for the reference configuration

of the ship. A number of terms of the

order of magnitude o(E2) are neglected on

the right-hand sides of formulae

(28).

However, it should be observed that, as a

result

of

assumptions

(6) ,

in

(28)

rotation tensor R can be replaced by the

time

independent

unit tensor

I.

This

shows

that,

within

the

adopted

approximation,

the hydrodynamic forces

and moments generated by pressure field

Pu on the ship are steady, and as stated

above their evaluation is not necessary

for determining the unsteady hydrodynainic

loads and ship motions.

Therefore the

unsteady resultant hydrodynamic force F

and moment rCG are composed of the

Froude-Krylov

forces

defined

by

expressions

(27),

_and of scattering force

F'

and

ruminent rCGS:

F=

_::

(29a)

r

= +

(29b)

where:

= R f

p5

NdS

(30a)

and:

rcus = R f p (L-L.0) x NdS (3 Ob)

For

further considerations

it

is

convenient to express force F5 and moment

in the following form:

F5=Rf

(31a)

= R

-

X

f)

with the force

=

f, ps N dS

and moment

Vo=f5PSxidS

(3ld)

defined in the reference configuration of

the ship.

2.3

_{The scattering potential and loads;}

the method of modal potentials

It

appears

from

the

preceding

discussion that ïn the theoretical model

applied

here,

the

determination

of

unsteady hydrodynamic loads depends in a

crucial way on the ability to evaluate

the

scattering

potential

and

corresponding

scattering

loads.

The

method used for solving the scattering

problem

is

the

method

of

modal

potentials, also developed and applied

under

the

name

of

equivalent

motion

method,

(Pawlowski,

1982),

(Pawlowski,

Bass

and

Grochowalski,

1988)

and

(Pawlowski and Wishahy, 1987). According

to the method the unknown scattering

potential

is approximated by a finite

series

of

scattering

potentials

,,

i=l,2,...,m,

in such a manner that the

error

of

approximation

of

the

impermeability condition on the ship's

hull

is

minimized.

Potentials

,

individually

satisfy

impermeability

conditions defined by their corresponding

normal

shape

functions

'i'M'

i=l,2,..m,

which are defined on the hull surface,

independently

of

its

orientation

and

location in space.

In

accordance

with

the

modal

potentials method, (Pawlowski, 1982) and

(Pawlowski, Bass and Grochowalski, 1988),

the normal velocity distribution v,,(x) on

the

instantaneous

wetted

surface

S,

defined by equation (9b), is approximated

by:

a

_{f31 T}

_(Th,

on s

_(32a)

Time

dependent

modal

amplitudes

i=l,2 ...in,

are

derived

from

the

condition that the residual of relation

(32a) be minimized, in the least squares

sense,

by

minimizing

the quantity>Q

defined as:

irr

d2 = r

_{(v12() -}

E

1ÍNj ()]2dS

JS,,(C) ₁₌₁

The minimization leads to the set

of

normal equations:

m

Z A1 f3

B,

for j = 1,2,.., m

1=1

and: B

=

f

'F

v1

dS,

for j

= 1,2,..

,m (32e)

from which the instantaneous values of the modal amplitudes are determined. Normal shape functions 'i'M must be chosen in such a manner that the main determinant of the system of equations

(32c) does not vanish. This means that the functions must be linearly independent on S,(fl

Starting from_equation (9a) and using the mapping X1-X1(x,t), mentioned

earlier (see Figure 1), between points in the instantaneous wetted surface S and points X1 in the reference wetted

surface S0, equation (32a) is reduced to (Pawlowski, in preparation):

m

(X1) (X1)

Z

'INi ()

11

on S

(33a) where quantities of the order of

magnitude o(e2) are neglected on the left-hand side. The motion of the ship brings every point in the wetted surface to the instantaneous location in space from

its

location X(,t)

in

the reference

configuration of the ship. Since normal

shape functions are defined on the surface of the ship independently of the ship motion, they are effectively defined

in the reference configuration, and:

NÍ N1 (33b)

In general, points X(,t) and

X1(,t)

in

the reference configuration do not coincide, as shown in Figure 1. however the distance between them, X-X1, is of the order of magnitude O(e). The normal shape functions are assumed to satisfy the Lipschitz condition:

- Ni (XL) MIX-X11 (33c)

where M is a positive constant, for any two points X1 and X on the surface of the ship. The condition applies also to

points X and X1 in equations (33a) and (33b). From relations (9c) and (32a) it follows that ß are of the order of magnitude o() and from (33c) it is found that *M(X) - (X1) are of the order of

magnitude 0(6). consequently, neglecting on the right-hand side of formula (33a) quantities of the order of magnitude 0(62), _{the impermeability condition for}

the

scattering flow reduces to:

= fS(C) N2 'P

d s,

forj,i =

1,2,..,m

(32d) m

ti

(X) N (X) = _{I '1Ni (X),} on S0

i=1

(33d)

which applies on the reference wetted

surface. In the following it is assumed that S0 is bounded on the hull surface by the calm water waterline.

Taking advantage of the form of

impermeability

condition (33d), scattering potential is expressed by a finite series of scattering potentials :

m

c ₌ Z 41 (34a)

where each of potentials satisfies Laplace's equation and, on the basis of equations (l5a), (33d) and (24), the

impermeability condition:

= on S0

(34b)

and free surface condition:

[(_u)2+g-]j=o

on

x3=o

(34c) The potential which satisfies conditions (34b) and (34c), and the causality condition,

stating that the

flow disturbance generated by excitation 3(t) cannot precede the excitation in

time, is expressible by means of the convolution integral with respect to time:

=

t3(t)

+

fl

(t) z. (t-t) dt

(35a) which is analogous to the expression

introduced in (Cummins, 1962) for the velocity potential induced by a ship displacement. In the expression on the right-hand side of formula (35a), represents the instantaneous response potential satisfying Laplaces equation and the following impermeability and free surface conditions:

NV4i1

= N1 onS0 (35b)

iV1

= o,

on X3 = O (35c)

The instantaneous response potential is independent of time. Under the convolution integral, ic(t-r) represents the memory potential which determines the effect at time t of the excitation at time r. The memory potential satisfies Laplace's equation, the causality condition:

the impermeability and free surface conditions: V ic (t) = 0, , for t > O (3 5e) - +

g]

K1(t)

= (35f)

onX3 O

for

t>O

and the initial condition on the free

surface:

ic1 (t) =

-g-i;- i'i'

(35g)

onX3 =

O, at

t = O

The last relation links the memory potential with the excitation, through the free surface effect of the

instantaneous response potential.

Equations (34a) and (35) show that

by the method of modal potentials the

determination of the scattering potential is reduced to the determination in time of a finite number of modal amplitudes $,, through the solution of the set of normal equations (32c) . Another necessary element constitutes the knowledge of the instantaneous response and memory potentials which, however, depend only on

the geometry and forward speed of the

ship, and on the choice of normal shape functions. Therefore these potentials can be obtained independently of any particular wave conditions and history of ship motion. It should also be observed that modal amplitudes constitute

non-linear functions of ship displacements and velocities, and non-linear functionals of the velocity potential of the oncoming wave. As a result, formulae (34a) and (35a) provide a non-linear expression for the scattering potential, which may be named quasi-linear because of its linear appearance. Th resultant scattering force and moment are obtained from equations (31) by using the quasi-linear representation of the scattering potential, equations (34a) and (35a), in conjunction with formulae (26d). The components of the force and moment, in the X system of reference, can be considered as generalized forces f, with f denoting the j component of force f, for j=1,2,3, and the j-3 component of moment -y,,, for j=4,5,6. Using that

notation: in = Z f71 1=1 (36a) -p

f (

-

u__)

4 (,

£

i

dS

(36b) for i=l,2,.., m, _{where f} is the j th

generalized scattering force induced by the scattering excitation in mode i. The application of a generalized form of Stokes' theorem, (Salvesen, Tuck and Faltinsen, 1970) and (Pawlowski, 1982)

together with the assumptions specified earlier, leads to:

= fsO 4i. (,

xThdS

uf 4

(0, ¡1xTh dS (36c)

_uf1 [

x dÄ, Xx (

x d5J}

The last integral on the right-hand side of formula (36c) is taken along the calm water waterline in the clockwise direction relative to the vertical axis.

The insertion in (36c) of modal scattering potential determined by equation (35a) gives the comprehensive formula for generalized forces

=

(t)

o)

-

(t)

U(

-

(o), X 11j

-

)

- f

_{A

(t)

(t-t),

(t-r)]

+(t)

CIL (t-r) x ¡ (t-r) -

K0) (t-r) ]

}dr (36d)

for j=l,2, . . 6 and i=1,2, . . in, with denoting the time derivative of _1,. The vectors of added masses ji,°')

nd memry forges

and moments (K(t),

K'(t) ,

!°>' (t)) ,

which occur in formula (36d) are defined as follows:

=

(,

x _Th dS (3 6e) O(e) =

Pf

(36f) x d, L x 1k1 ( t) ,

k° (

t) ]

- -

(36g) =

pf

K1(t) (N, Xx N) dS for J=l,2,..,6, with:

[k1'

(t) ,

k°'

(t) ].=

f

(3 6h) x d, x ( x

Formula

(36d)

provides

a

quasi-linear

expression of modal scattering forces.

Relations

between

the

added

nass

and

memory force vectors introduced above,

arid well known frequency dependent added

mass

and

damping

coefficients,

are

presented in Appendix 2.

2.4

The

implemetatiofl

of

the

theoretical model

The

theoretical

model

presented

above gives a representation of unsteady

hydrodynamic

loads

(i.e.

loads

corresponding

to

the

ideal

fluid

assumption) exerted on a ship advancing

in

steep waves.

However,

an

adequate

prediction

of

unsteady

ship

motion

depends

as

well

on

the

ability

to

estimate properly unsteady fluid loads

which result from viscous effects in the

flow.

This

requirement

applies

in

particular to the viscous damping moment

in the roll mode of motion, and to the

force and moment respectively in the sway

and

yaw

modes

of

motion,

which

are

related

to

lift

and

drag

phenomena

generated on the hull, skeg and control

surfaces. The viscous damping of roll is

well known to be of crucial importance

for a good prediction of roll notion. The

sway force and yaw moment appear to be

equally

important

because

of

their

influence

on

the

prediction

of

the

instantaneous location of the ship

in

waves.

The

above

theoretical

considerations,

supported

by

computational experience,

indicate that

for the non-linear ship motion model a

successful prediction of fluid loads and

ship motions is coupled strongly with the

ability

to

predict

with

sufficient

accuracy the instantaneous position of

the ship in waves.

At present those real fluid effects

can only be included in the theoretical

model

by

means

of

semi-empirical

formulae, such as formulae presented ïn

(Himeno,

1981)

and

(Crane,

Eda

and

Landsburg, 1989), which are applied here.

This requires the implicit description of

fluid kinematics, in terms of ship sway

linear velocity and roll and yaw angular

velocities, used in the semi-empirical

formulae, to be matched with the explicit

description,

in

terms

of

nodal

amplitudes,

3, and corresponding normal

shape

functions,

4'M'

used

in

the

theoretical

model.

The

matching

is

achieved

by

interpreting

some

of

the

modal amplitudes, for which normal shape

functions are appropriately defined, as

corresponding to the local values of the

ship velocities.

When necessary those

velocities are averaged for use in the

semi-empirical formulae.

A proper selection of normal shape

functions is also necessary to obtain an

adequate accuracy of the representation

of the scattering flow and loads. From

the above considerations it follows that

two

restrictions

are

imposed

on

the

choice of the normal shape functions, the

requirement of their linear independence

on the wetted surface S(fl,

and the

requirement of a local representation, by

some

of

the functions,

of

the normal

velocity of the wetted surface due to

rigid body motions in the sway and roll

modes.

It

is

therefore

convenient

to

define the normal shape functions as a

doubly indexed set:

=

,with i = n2 (k-1) + i

(37a)

for

i=l,2,..,m,

k=l,2,..,n1,

and

l=1,2,..,n2. The normal shape functions

with index k are chosen to be identically

equal

to

zero

on

a

surface

S which

contains

all

possible wetted surfaces

S(), with the exception of a section SSk

of

S.

The

sections

_65k

are

mutually

exclusive and cover the whole of S:

S =

(37b)

In

order

to

apply

the

semi-empirical

formulae for viscous loads, sections &Sk

are

defined

as

parts

of

surface

S

contained

between

suitably

chosen

consecutive

transverse

cross-sections

spaced

along

the

hull.

Computational

experience

(Pawlowski,

Bass

and

Grochowalski, 1988), (Pawlowski and Bass,

1990,a) and (Pawlowski and Bass, l990,b),

indicates

that

the

following

set

of

sectional

normal

shape

functions

4',,,

l=l,2,..,5

is

usually

sufficient

for

computations of ship motions in waves:

= N1, = N2, = N3,

(37c)

= X2N3 - X3N2, _Nk5 = X3N2

for k=l,2,..,n1

where N., i=l,2,3, denote

components of N. For the application of

the semi-empirical formulae, the values

of

modal

amplitudes

and

ß

which

correspond to normal shape functions 'I',,

and

are interpreted respectively as

the local (sectional) values of sway and

roll velocities.

The weak scatterer assumption which

is

adopted

as

the

basis

of

the

hydrodynamic model implies that the ship

moves compliantly with the oncoming wave.

This may not occur if in the presence of

ari oncoming wave the ship is restrained

or forced to move also by other external

forces. Then quantities neglected in the

model

as

being

of

sufficiently

small

order

of

magnitude,

O(e2),

may become

significant. Taking full account of such

effects requires at least the development

of a consistent theoretical model of the

third order, i.e. a model which includes

all terms larger and of the order of

magnitude Q(3) Such a development is

beyond

the

scope

of

the

present

discussion. However,

_it

_{is possible to}

include in the present second order model

some of

those effects

in

an averaged

form.

Taking into consideration an individual modal component v of the normal velocity v, determined according to the approximation (32a):

= t3 (38a)

its modal amplitude 3, may be obtained

from the normal equation:

,

f

WdS

= f

vY dS

(38b)

At the adopted level of consistency, in the computation of the scattering potential the left-hand side of equation

(38b) is approximated by:

f

T

dS 13

L

'F dS (38c) with the error of the approximation being of the order of magnitude o(2). Instead of using (38c), can be replaced by ß' which compensates for the error:

= f

V1TF1 dS

(39d)

This leads to the corrected values of

modal amplitudes

ß'

defined by the relations:

=

_(f3

ds) /

dS

(39e)

for i=l,2,..,m. The corrected modal amplitudes are used in the model without violating its consistency. The effects they introduce are significant in the modelling of the interaction of a

restrained ship with steep waves. 3.0 EXAMPLES OF COMPUTATIONS

3.3.

Computation of modal

scattering potentials

Owing to its underlying simplicity, the above described theoretical model can be implemented, with appropriate adaptation, using any of the currently known methods for the solution of linear radiation problems. Examples of such applications are provided in (Pawlowski, Bass and Grochowaiski, 1988), (Pawlowski and Wishahy, 1987) and (Pawlowski and

Bass, 1990,a). The results obviously depend on the applied method. The methods can be classified crudely as two-dimensional and three-two-dimensional (the radiation problem is formulated and

solved in two dimensional or in three

dimensional space), and as frequency domain and time domain methods (the problem is formulated and solved assuming

a steady state, harmonic in time excitation, or an arbitrary transient excitation). Another classification corresponds to a variety of numerical methods available to solve the problem in a given formulation. The best known are panel (boundary element) methods based on

the use of Green's functions which satisfy a linear free surface condition (either in the frequency or in the time

domain) . _{Boundary element methods which} employ Rankine source distributions, and finite element methods are also available. The third type cf classification depends on whether or not the forward speed effect is included in the free surface condition. That classification essentially does not apply to three dimensional, time domain Green's function formulations.

Mere, the choïce is made to use a three dimensional, frequency domain

method which employs the free surface

Green's function without the forward speed effect. Such a selection is a

result of compromise between several conflicting demands. Although the

required amount of data preparation and computation grows significantly as one opts for a three dimensional instead of two _{dimensional formulation, the} efficiency of the three dimensional computations has increased markedly and there is no compelling practical need to forsake the modelling of three dimensional flow effects. The robustness achieved by some (so called second generation) of the three dimensional algorithms of the chosen type provides an incentive. An additional advantage follows from the computation in the frequency domain in which every modal scattering problem is solved for a series of frequencies, and subsequently the resulting series of solutions is transformed, (Pawlowski and Bass, 1990, a), into a required time domain representation. Since the solutions at any two frequencies are independent, possible convergence and irregular frequency problems are much easier to identify and correct, than in corresponding time domain solutions. The main disadvantage of the chosen method for the present application is that it does not include the forward speed effect in the free surface condition. However, similar methods which include this effect have not yet achieved the same level of reliability and efficiency, due to the much more difficult problem of evaluating

the frequency domain Green's function

they employ. That disadvantage is to a degree offset by the present application,

since in heavy seas the ship's speed is usually reduced, and therefore prominent forward speed effects may not occur. The

computer program used to generate the

series of frequency domain scattering added mass and damping coefficients, (see Appendix 2), which are subsequently transformed into the required time domain quantities, is M-WANIT, a modified version of WAMIT, (Newman and Sclavounos,

1988) . In addition, in the present computations the waterline integrals (36f) and (36h) were neglected as

providing

sufficiently

small

contributions, although this may not be justified in the case of the fishing vessel discussed below.

SERIES 60 TESTS. HEADING 60 deg.. Fn=0

3.2 Captivs and fr.. running simulations for a series 60 model

Unsteady fluid load and ship motion simulations, obtained by a computer program implementation of the non-linear time domain model described above, are presented here for two very dissimilar vessels.

One is the Series

60, block coefficient 0.60,

hull which has been

extensively tested in regular oblique waves of small amplitude, (Lewis and Numata, 1960) and (Chey, 1963). The other vessel is a small stern trawler, tested

in periodic, steep (close to breaking and breaking) waves, (Grochowalski, 1989).

Experimental results from the two test series were applied previously in comparisons with numerical simulation

results generated with the use of

an earlier version of the present hydrodynamic model, (Pawlowski, Bass and Grochowalski, 1988)

Some of the main particulars of the Series 60 model are given in Table 1. For the purpose of the computations the hull surface was discretized to the deck level into 1828 quadrilateral panels (1468

LU LU o o LU o I--J LU -J LU o 8 0 0.4 0.3 0.2 0.1 0.0 0.8 0.6 0.4 0.2 0.0 0 0.5 1.0 1.5 2.0 2.5 SERIES 60

NONDIMENSIONAL SWAY FORCE

i

o

I

o 00 TESTS. HEADING 60 0.1 0.075 0.05 LU 0.025 z X/ L o 0.0 0 o al o 0.1 o -J 0.075 -J LU 0.05 8 0.025 XIL 0.0 deg 0.5 1.0 1.5 2.0 2.5 .

Fn0.I

NONDIMENSIONAL PITCH MOMENT

o Q

O

),IL

00 0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL HEAVE FORCE

- u O

o

Q

0 0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL YAW MOMENT

o o

o

e

0.1

NONOIMENSIONAL SWA FORCE NONDIMENSIONAL PITCH MOMENT

o 0.075 O D LU 0.05 2 -J XII. z o 0.025 0.0 I/L 0 0 0.5 1.0 1.5 2.0 2.5 o _0.100 0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL HEAVE FORCE

LU

o NONDIMENSIONAL YAW MOMENT

e 0.075 o 0.05 LU J 0.025 XL 0.0 XII. c.o 0.5 1.0 1.5 2.0 2.5 00 0.5 1.0 1.5 2.0 2.5

Fig.2 comparison of diffraction load amplification factors, experimental values from (Chey, 1963),

D experiment, computation. 0.4 0.3 z 0.2 LU 0.2 LL LU o 0.0 o LU 0.8 0.6 0.4 0.2 8 0.0

Fig. 3

SERIES 60 TESTS. HEADING 60 deg., Fn=0.2

Comparison of diffraction load amplification factors, experimental values from (Chey, 1963),

C experiment, A computation.

panels below the waterline, including the waves of small amplitude. For each rudder), grouped into 48 sections. A heading, the model was aligned so as to

series of time domain simulations was

make its heading and the direction of

performed for the restrained vessel to advance coincide. The rudder was fixed match the experiments, reported in (Chey,

1963), carried out in the square

seakeeping tank of the Davidson Table i Some of the main particulars of Laboratory, and aimed at determining the the Series 60 hull form.

wave-excited forces and moments about CG,

in regi.ilar oblique waves of small L 1.524 n

amplitude. In the experiments the ship

LIB

7.50

model was restrained in all six degrees

B/T

2.50

of freedom and towed with constant C8 0.60

speeds, at various headings in regular

0.0 00 AL TESTS, HEADING 60 A/L 0.5 1.0 1.5 2.0 2.5 SERIES 60 00 0.5 1.0 1.5 2.0 2.5 deg.. Fn=O.3 0» 0.1

NONDIMENSIONAL SWAY FORCE NONDIMENSIONAL PITCH MOMENT

0.3 0.075 0.2 J

-

0.05 a D X X O 0.1 w D 0.025 ç) X o z

-

0.0 AL 0.0 AIL 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5 o 00 0 C C

0.8 NONDIMENSIONAL HEAVE FORCE w_C

=

0.1 NONDIMENSIONAL YAW MOMENT

' _0.6 0.075 X C 2 O.A a a 0.05 J o 0.2 - 0.025 u Q 0.0 0.0 AIL c.S :.o 1.S 2.0 2.5 00 00 0.5 1.0 1.5 2.0 2.5 0.0 o AIL o 0.0₀ AIL 0 0.5 1.0 1.5 1.0 2.5 0 0.5 1.0 .5 2.0 2.5 0.8 0.1

NONDtMEMSIONAL HEAVE FOR _w NONDIMENSIONAL YAW MOMENT

C 0.6 _0.075 Q ¿ O 0.A 2 Q = 0.05 a o 0.A 0.1

NONDIMENSIONAL SWAY FORCE NONDIMENSIONAL PITCH MOMENT

0.3 0.075 0.2 Q a X 0.05 O a o 0.0 2 0.025 X C I-z 0.2 0.0

0.0 00 0.8 0.6 0.A 0.2 0.0 00 a AL 0.0 AIL 1.0 1.5 2.0 2.5 Q 0 0.5 1.0 1.5 2.0 2.

Comparison of diffraction load amplification factors, experimental values from (Chey, 1963),

O experiment, computation.

0.0

00 0.5

Fig.4

amidships on the model, throughout the experiment, and a free rotating propeller was installed. The model was tested without bilge keels.

Sway and heave

forces,and pitch and yaw moments induced on the model were measured by a

dynamometer by means of which the model was attached to the towing carriage. The oscillatory forces were presented in terms of non-dimensional double amplitudes. In the computations, the

waves were

simulated using Airy wave potential, with the same wave height as used in the experiments. Double force and moment amplitudes were read from the

lu I-z o o u' p. "J -I 0.05 0.025 0.025 0.0 0.5 1.0 1.5 2.0 :1: o

SERIES 60 TESTS. HEADING 20 dog.. Fn=0.l 0.0

0 0 0.5 1.0 1.5 2.0 2.5

AL

2.5

NONDIMENSTONAL PITCH MOMENT

o

0.0

00 0.5 1.0 1.5 2.0 2.5

0.1 NONDIMENSIONAL YAW MOMENT

a

9

u

A/L

X'L

computed records just after the steady state of response was achieved.

A comparison of non-dimensional force and moment amplification factors derived from the computations, with the ones reported in (Chey, 1963) for the headings of 60 and 120 degrees, is presented in Figures 2 to 5. The

comparison of the results of the time

domain simulation with the experimental values is good with the exception of the pitch moment for which a growing

discrepancy is observed as the forward speed increases from Froude number 0.2 to 0.3 and as the heading decreases from 120

'SI E u. o lu 0.3 0.2. 0.1 0.0 0 0.8 0.6 a

NONOTMENSTONAL SWAY FORCE

J

8

0 0.5 1.0 1.5 2.0 ¿.5

NONDIMENSIONAL HEAVE FORCE

SERIES 60 TESTS. HEADING 120 deg.. Fn0

0.,

NONOIMENSIONAL SWAY FORCE

0.1

NONDIMENSIONAL PITCH MOMENT

0.3 o a 0.075

I

0.2 0.05 9 0.025 0.1 I-z 0.075

.

0.05 lu 0.025 u' -J A E 0.2 0.5 1.0 1.5 2.0 2.5 ! >.L 0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL HEAVE FORCE

a

AL

NONDIMENSTONAL YAW MOMENT

u o o 0.i 0.1 0.075 0.05

0.2 0.1 L) o = E 0. 0.5 0.5 OA 0.2 0.0 o

Fig. 5

D

I

o 8 8

I

NONDIMENSIONAL SWAY FORCE

0.0

O _{o 0 0.5 1.0} 1.5 2.0 2.5 0.8

F4ONDIMENSIONAL HEAVE FORCE

o

+ V/I.

0 0.5 1.0 1.5 2.0 2.5

to 60 degrees. These discrepancies nay be attributed to the method used to compute the added mass and damping coefficients, which, as indicated in (Chang, 1977) and

(Inglis and Price, 1982), with growing forward speed fails increasingly to

predict correctly pitch added mass and damping coefficients at low frequencies of excitation. For the discussed regimes of operation the frequency of encounter decreases with the heading angle and with increasing wavelength. With increasing forward speed the frequency of encounter decreases at the 60 degrees heading, and

SERIES 60 TESTS. HEADING 120 deg.. Fn-0.2

0.1 0. 075 0.05 = z 0.0 0.05 = E 0.025 00 0.5 1.0 1.5 2.0 0.1 0.075 0.025

SERIES 60 TESTS, HEADING 120 deg.. En0.3

0.1

NONOIMENSIONAL YAW MOMENT

Comparison of diffraction load amplification experimental values from (Chey, 1963)

D experiment, computation.

factors,

L

increases at the 120 degrees heading. The difficulty in the proper prediction of pitch can be removed, at least to some extent, by the proper inclusion of the forward speed effect in the free surface condition. Otherwise it imposes a

limitation on the range of forward speed possible to model reliably for a given heading.

In Figure 6 results of time domain simulations of the vessel's motions in a regular wave train of small amplitude, at a heading of 60 degrees, are compared with the corresponding experimental

0.3 0.2 = 0.1 L) E 0.0 o _0.0 -J 0.8 0.6 2 0. 0.2 0.0

NONDIMENSIONAL SWAY FORCE

D o D S VIL V/I. = I-z 'JI O UI O = E 0.075 0.05 0.025 0.0 0.1 0.075 0.05 0.025 0.0 0 I

NONDIMENSTONAL PITCH MOMENT

D

o

X t.

V/L

0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL HEAVE FORCE

2 S 2 o 00 0.5

I

1.0 1.5 2.0 2.5

NONDIMENSIONAL YAW MOMENT

I

00 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 .5 .0 ¿.5

0.0 X 'L

0 0 0.5 1.0 1.5 2.0 2.5

NONDIMENSIONAL PITCH MOMENT

a D

005 0.1 0.15 0.2 0.25 03

values obtained from free running model tests reported in (Lewis and Numata,

1960). The tests were performed in the

same Davidson Laboratory tank as the captive model tests described above, and

the same model was used

(Chey, 1963). However, the model was self-propelled and steered by an automatic pilot system. It

was completely free to move in all six degrees of freedom, but was attached to a motion-recording apparatus designed to follow the motion of the model with minimum interference. In the simulations the vessel is propelled by a longitudinal force adjusted

in time to maintain

a

required mean forward speed. In addition rudder forces are evaluated, which together with an autopilot loop serve to provide a realistic modelling of yaw motion in waves, and give coupling effects with roll. For each shown forward

speed the simulation was run over 24

encounter periods and the amplitudes of

motion were averaged over the last

15 periods.

In general, the agreement between the values derived from the simulation and the experimental data is good with the exception of roll motion. Greatest discrepancies are observed at Froude number O for all modes of motion. This is

00 _0.05 0,1 015 0.2 025 0.3

a consequence of the difficulty to control the mean heading of the vessel at zero forward speed, which is experienced in the simulation, and was also a factor during the model tests. The problem extends to low forward speeds, at which the control of the mean yaw angle (to be distinguished from the heading angle) affects the motions, in particular in the roll mode.

The roll amplitudes are largely overpredicted, although the simulated and experimental values follow closely the same qualitative pattern, with the exception of the experimental point at Froude number 0.3. The overprediction results probably from a combined effect of an underestimation of roll damping in the numerical model, the influence of the mean yaw angle, the contribution to the roll moment provided by the rudder action

which may

be quite different in the simulation in comparison with experiment, and the influence of the propeller in the experiments. In addition, some of the experimental runs in quartering seas showed variations in roll amplitude, for which changes in model speed and heading were identified as possible causes. A more detailed analysis of the discrepancy could be carried out only by comparing SERIES 60.}{EADING-60 dqWAVELRNGTF1tEN0114-I.0.FREE RUNNING SERIES 60.5{EADING-60 &gWAVELENGTh/LENGTH-I O.FREE RUNNING

07 6 06 10 s A. 0s 10 4 LO > 04 2 _{3 02} C 10 0.1

SERIES 60.HEADING6O dcg.WAVELENOTHjLENGTH- 10,FREE RUNNING ₀₈SERIES 60.1-lEADING =60 dqWAVELENGTHJLENGTh.LO,FREE RUNNING

0.7 06

8

5

04 07 3 s 06' > 0.2

-

05 040 01 0.5 01 015 02 025 03 S.S 01 015 0.2 0,25 03

FR01100. NUMBER FROUDE NUMBER

FROUDE NUMBER FROUDE NUMBER

Fig.6 Comparison of non-dimensional motion amplitudes, experimental values from (Lewis and Numata, 1960)

z

3

0.5 1 1.5

TIME ()

2 2.0

the simulated and experimental motion records.

It should be noted that the

same kind of motion characteristics related to the control of the model in oblique seas, as observed during the experiments, that is the occurrence of mean yaw angle and lateral drift, and gradual variations in yaw, sway and surge, were present in the simulations. Another notable feature of the comparison

is the gradual deterioration, with increasing forward speed, of the agreement between the simulation and experiment in the pitch mode. This may be caused by the already explained inadequate modelling of the scattering pitch moments at higher forward speeds.

z O 3 >. 0.5 I LS TIME (L)

Fig.7 comparison of motions and loads in the partly captive condition, steady free motion sìmulation. Experiment described

in (GrochOWaiSki, 1989), * experiment, - computation.

2 2.5

3.3 Partly captive and free running simulation for a stern trawler

Some of the main particulars of the hull form of the stern trawler are provided in Table 2. For the computations the surface of the vessel, including the bulwark, weather deck and superstructure, and the rudder, was discretized into 1420 quadrilateral panels grouped into 47 sections (766 panels below the waterline, including the rudder). Computations were carried out for partly captive and free running regimes of motion in steep waves, which correspond to chosen fragments of

two experimental test runs performed

TEAWLERPARTLY CAPTIVE. STEADY MOTION TRAWLER.PARTLY CAPTIVE. STEADY MOTION

150 15 IS 50- z 10 .5o. .5 -150 10 0.3 1 1.5 2 2.5 0.3 1 LS 2 2.5 TIME () TIME (L)

TRA\ER.?ARTLY CAPTIVE. STEADY MOTION TLAWLEE.PARUY CAYS1VE, STEADY MOTION

150-s

30-a .10 -1010 OES 1 LS

i,

03 1 Ii 2 0.5

¡

as o -as TIME () TIME (n)